| L(s) = 1 | − 2·9-s + 24·17-s − 4·25-s − 24·41-s − 4·49-s + 8·73-s + 3·81-s + 24·89-s − 8·97-s − 24·113-s + 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 48·153-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 5.82·17-s − 4/5·25-s − 3.74·41-s − 4/7·49-s + 0.936·73-s + 1/3·81-s + 2.54·89-s − 0.812·97-s − 2.25·113-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 3.88·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.661315477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.661315477\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.201938143900970063825023534401, −9.095635300961643552420937018514, −8.319747243197069991378387465816, −8.314869136830471831185090861616, −8.186881124766838583071624871498, −7.927093271330810909723914638442, −7.52074758091274109258973991389, −7.49608297428456407648782546550, −7.08786237725335792325890154552, −6.61017659790386139802871915396, −6.49946332206836102982452761761, −5.90217453832004117775645133580, −5.73225674221684299987607723230, −5.59151821386866085061786196819, −5.24352458317283843766027312129, −4.96815686767717604150997629660, −4.78591559504293675955562833554, −3.93684381690113906940585843183, −3.62548604982906873751766525295, −3.45352217451139096966025524880, −3.10614619104068965326720920865, −2.94859128859384174501581603221, −2.00624925349395138781280071069, −1.53191267870496635938549567217, −0.941758406426872705971917835081,
0.941758406426872705971917835081, 1.53191267870496635938549567217, 2.00624925349395138781280071069, 2.94859128859384174501581603221, 3.10614619104068965326720920865, 3.45352217451139096966025524880, 3.62548604982906873751766525295, 3.93684381690113906940585843183, 4.78591559504293675955562833554, 4.96815686767717604150997629660, 5.24352458317283843766027312129, 5.59151821386866085061786196819, 5.73225674221684299987607723230, 5.90217453832004117775645133580, 6.49946332206836102982452761761, 6.61017659790386139802871915396, 7.08786237725335792325890154552, 7.49608297428456407648782546550, 7.52074758091274109258973991389, 7.927093271330810909723914638442, 8.186881124766838583071624871498, 8.314869136830471831185090861616, 8.319747243197069991378387465816, 9.095635300961643552420937018514, 9.201938143900970063825023534401
